A257180 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(-24 - 30\,x - 12\,x^{2} - 10\,x^{3} - 4\,x^{4} - 12\,n - 15\,n x - 19\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} - 6\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 2\,x^{2} + 4\,x^{3} + 2\,x^{4} + 2\,x^{5} - 3\,n x + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(10 + 17\,x + 6\,x^{2} + 2\,x^{3} + 7\,n + 5\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 33 + 42\,x + 9\,x^{2}\)

\(P_{2}(x) = 168 + 423\,x + 222\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	33	42	9								84
2	168	423	222	27							840
3	807	3546	3588	1050	81						9072
4	3714	26283	45624	25290	4686	243					105840
5	16557	178482	491943	462180	160467	20202	729				1330560
6	72060	1136391	4704822	6952605	4056816	952497	85182	2187			17962560
7	307923	6891666	41122746	90478374	82513284	32380110	5404686	353850	6561		259459200
8	1297302	40256379	335424048	1053890124	1422867852	869013978	241737216	29710332	1454766	19683	3995671680

All 15 rows

Technical details

Node id: n-5f11ae285e99a7ba · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-dc8c0af61de5a33e

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       33,         42,           9;
   2:      168,        423,         222,           27;
   3:      807,       3546,        3588,         1050,           81;
   4:     3714,      26283,       45624,        25290,         4686,          243;
   5:    16557,     178482,      491943,       462180,       160467,        20202,          729;
   6:    72060,    1136391,     4704822,      6952605,      4056816,       952497,        85182,        2187;
   7:   307923,    6891666,    41122746,     90478374,     82513284,     32380110,      5404686,      353850,        6561;
   8:  1297302,   40256379,   335424048,   1053890124,   1422867852,    869013978,    241737216,    29710332,     1454766,     19683;
   9:  5405721,  228348090,  2591790117,  11258029128,  21579387858,  19441213260,   8395158450,  1718768808,   159620757,   5937162,    59049;
  10: 22331472, 1265481543, 19182923454, 112276473999, 295646660136, 376774038606, 239481290580, 76019039766, 11782059096, 843206859, 24102942, 177147;

%F Formula

The row polynomials satisfy (n+1+2*x)*P(n,x) = -(12*n+24+(6*n^2+15*n+30)*x+(n^3+4*n^2+19*n+12)*x^2+(3*n^2+8*n+10)*x^3+(2*n+4)*x^4)*P(n-2,x) - ((3*n+6)*x+(n^2+2)*x^2-(n^2+4)*x^3-(3*n+2)*x^4-2*x^5)*d/dx P(n-2,x) + (7*n+10+(2*n^2+5*n+17)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) + ((n+1)*x-(n-1)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 33 + 42*x + 9*x^2, P(2,x) = 168 + 423*x + 222*x^2 + 27*x^3.

%t Mathematica

P[1, x_] := 33 + 42*x + 9*x^2;
P[2, x_] := 168 + 423*x + 222*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 24 + (6*n^2 + 15*n + 30)*x + (n^3 + 4*n^2 + 19*n + 12)*x^2 + (3*n^2 + 8*n + 10)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((3*n + 6)*x + (n^2 + 2)*x^2 - (n^2 + 4)*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (7*n + 10 + (2*n^2 + 5*n + 17)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 33 + 42*x + 9*x^2; v[2] = 168 + 423*x + 222*x^2 + 27*x^3; for(n = 3, N, my(i = n); v[i] = (-(12*n + 24 + (6*n^2 + 15*n + 30)*x + (n^3 + 4*n^2 + 19*n + 12)*x^2 + (3*n^2 + 8*n + 10)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((3*n + 6)*x + (n^2 + 2)*x^2 - (n^2 + 4)*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (7*n + 10 + (2*n^2 + 5*n + 17)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
    
    ↑ Liu–Wang theorem (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.